In addition to the techniques of integration we have already seen, several other tools are widely available to assist with the process of integration. Among these tools are
integration tables , which are readily available in many books, including the appendices to this one. Also widely available are
computer algebra systems (CAS) , which are found on calculators and in many campus computer labs, and are free online.

Tables of integrals

Integration tables, if used in the right manner, can be a handy way either to evaluate or check an integral quickly. Keep in mind that when using a table to check an answer, it is possible for two completely correct solutions to look very different. For example, in
Trigonometric Substitution , we found that, by using the substitution
x=tanθ, we can arrive at

∫dx1+x2=ln(x+x2+1)+C.

However, using
x=sinhθ, we obtained a different solution—namely,

∫dx1+x2=sinh−1x+C.

We later showed algebraically that the two solutions are equivalent. That is, we showed that
sinh−1x=ln(x+x2+1). In this case, the two antiderivatives that we found were actually equal. This need not be the case. However, as long as the difference in the two antiderivatives is a constant, they are equivalent.

Using a formula from a table to evaluate an integral

Use the table formula

∫a2−u2u2du=−a2−u2u−sin−1ua+C

to evaluate
∫16−e2xexdx.

If we look at integration tables, we see that several formulas contain expressions of the form
a2−u2. This expression is actually similar to
16−e2x, where
a=4 and
u=ex. Keep in mind that we must also have
du=ex. Multiplying the numerator and the denominator of the given integral by
ex should help to put this integral in a useful form. Thus, we now have

∫16−e2xexdx=∫16−e2xe2xexdx.

Substituting
u=ex and
du=ex produces
∫a2−u2u2du. From the integration table (#88 in
Appendix A ),

The region bounded between the curve
y=11+cosx,0.3≤x≤1.1, and the
x -axis is revolved about the
x -axis to generate a solid. Use a table of integrals to find the volume of the solid generated. (Round the answer to two decimal places.)

fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.

Tarell

what is the actual application of fullerenes nowadays?

Damian

That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.

Tarell

Join the discussion...

what is the Synthesis, properties,and applications of carbon nano chemistry

Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.

Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?

You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?